
// Make newform 1200.3.c.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1200_3_c_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [81, 0, -17, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -8, 0, 1], [-9, 0, 1, 0]];
    Rf_basisdens := [1, 1, 9, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();"
function MakeCharacter_1200_c()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [2, 2, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_Hecke();"
function MakeCharacter_1200_c_Hecke(Kf)
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, -8, 0], [3, 0, 0, 6], [0, 0, -2, 0], [0, -6, 3, 0], [11, 0, 0, 0], [0, 12, -6, 0], [6, 0, 0, 12], [46, 0, 0, 0], [0, 0, -16, 0], [-9, 0, 0, -18], [0, 0, 62, 0], [0, 12, -6, 0], [0, 12, -6, 0], [12, 0, 0, 24], [-16, 0, 0, 0], [0, 0, -113, 0], [-18, 0, 0, -36], [0, 0, -101, 0], [68, 0, 0, 0], [0, -6, 3, 0], [9, 0, 0, 18], [0, 0, -22, 0], [24, 0, 0, 48], [0, 0, 26, 0], [0, 6, -3, 0], [-176, 0, 0, 0], [0, 42, -21, 0], [0, 0, 106, 0], [-12, 0, 0, -24], [0, 18, -9, 0], [-127, 0, 0, 0], [18, 0, 0, 36], [-164, 0, 0, 0], [0, 0, 38, 0], [0, 0, 47, 0], [0, 60, -30, 0], [0, -24, 12, 0], [3, 0, 0, 6], [-106, 0, 0, 0], [42, 0, 0, 84], [0, 0, 193, 0], [0, -12, 6, 0], [-4, 0, 0, 0], [-209, 0, 0, 0], [0, 0, -148, 0], [0, 0, 0, 0], [-116, 0, 0, 0], [0, -72, 36, 0], [-30, 0, 0, -60], [59, 0, 0, 0], [21, 0, 0, 42], [0, -168, 84, 0], [0, -84, 42, 0], [-60, 0, 0, -120], [178, 0, 0, 0], [0, 0, -202, 0], [-12, 0, 0, -24], [0, 0, 197, 0], [0, 144, -72, 0], [0, 0, 127, 0], [-48, 0, 0, -96], [0, 0, 58, 0], [0, 96, -48, 0], [-257, 0, 0, 0], [0, 0, -127, 0], [0, -126, 63, 0], [292, 0, 0, 0], [0, -216, 108, 0], [-48, 0, 0, -96], [0, 0, -668, 0], [0, 0, -242, 0], [191, 0, 0, 0], [0, -96, 48, 0], [-102, 0, 0, -204], [0, 0, -772, 0], [-45, 0, 0, -90], [-701, 0, 0, 0], [-51, 0, 0, -102], [-148, 0, 0, 0], [36, 0, 0, 72], [0, 0, 463, 0], [-394, 0, 0, 0], [0, -222, 111, 0], [21, 0, 0, 42], [0, 0, 509, 0], [36, 0, 0, 72], [0, 0, -64, 0], [0, 216, -108, 0], [-42, 0, 0, -84], [0, 0, 382, 0], [120, 0, 0, 240], [-94, 0, 0, 0], [0, 264, -132, 0], [-54, 0, 0, -108], [-135, 0, 0, -270], [0, 0, 467, 0], [-976, 0, 0, 0], [0, 0, -779, 0], [0, -252, 126, 0], [0, 24, -12, 0], [165, 0, 0, 330], [286, 0, 0, 0], [0, 0, -487, 0], [0, -198, 99, 0], [0, 30, -15, 0], [108, 0, 0, 216], [-271, 0, 0, 0], [0, 0, -404, 0], [0, 0, 34, 0], [0, -144, 72, 0], [1046, 0, 0, 0], [106, 0, 0, 0], [-36, 0, 0, -72], [0, 0, 566, 0], [0, 168, -84, 0], [0, 168, -84, 0], [93, 0, 0, 186], [-298, 0, 0, 0], [0, 0, 274, 0], [0, 336, -168, 0], [0, -42, 21, 0], [373, 0, 0, 0], [114, 0, 0, 228], [184, 0, 0, 0], [108, 0, 0, 216], [0, 0, -758, 0], [0, 0, -26, 0], [1298, 0, 0, 0], [0, -180, 90, 0], [478, 0, 0, 0], [0, 0, -1072, 0], [93, 0, 0, 186], [169, 0, 0, 0], [0, -156, 78, 0], [0, 0, -878, 0], [0, 276, -138, 0], [-36, 0, 0, -72], [106, 0, 0, 0], [-102, 0, 0, -204], [0, 0, -1348, 0], [0, -366, 183, 0], [1222, 0, 0, 0], [72, 0, 0, 144], [0, 0, -902, 0], [0, -282, 141, 0], [-577, 0, 0, 0], [0, -492, 246, 0], [0, 0, 488, 0], [-180, 0, 0, -360], [0, 0, -313, 0], [0, 384, -192, 0], [0, 0, -374, 0], [108, 0, 0, 216], [1286, 0, 0, 0], [-84, 0, 0, -168], [0, 0, -1597, 0], [-42, 0, 0, -84], [0, 120, -60, 0], [0, -210, 105, 0], [0, 0, 1576, 0], [195, 0, 0, 390], [0, -162, 81, 0], [0, 480, -240, 0], [676, 0, 0, 0], [0, 0, 1004, 0], [-881, 0, 0, 0], [0, -372, 186, 0], [-87, 0, 0, -174], [104, 0, 0, 0], [156, 0, 0, 312], [0, 0, 433, 0], [-1294, 0, 0, 0], [159, 0, 0, 318], [211, 0, 0, 0], [-60, 0, 0, -120], [0, 0, -1408, 0], [-26, 0, 0, 0], [0, 0, 1192, 0], [87, 0, 0, 174], [0, 0, -1112, 0], [0, -150, 75, 0], [0, 228, -114, 0], [-156, 0, 0, -312], [0, 0, 1088, 0], [0, 0, 437, 0], [1549, 0, 0, 0], [-6, 0, 0, -12], [0, 0, 493, 0], [0, -546, 273, 0], [2011, 0, 0, 0], [-18, 0, 0, -36], [0, 54, -27, 0], [0, 264, -132, 0], [17, 0, 0, 0], [0, 0, 1354, 0], [0, 210, -105, 0], [0, -120, 60, 0], [-282, 0, 0, -564], [148, 0, 0, 0], [0, 0, -1396, 0], [1147, 0, 0, 0], [-396, 0, 0, -792], [0, -780, 390, 0], [866, 0, 0, 0], [0, -552, 276, 0], [-144, 0, 0, -288], [-1214, 0, 0, 0], [0, 0, -682, 0], [-84, 0, 0, -168], [0, 0, 902, 0], [0, -306, 153, 0], [366, 0, 0, 732], [-1003, 0, 0, 0], [0, 0, 1222, 0], [453, 0, 0, 906], [0, -192, 96, 0], [0, -504, 252, 0], [1034, 0, 0, 0], [86, 0, 0, 0], [111, 0, 0, 222], [0, 0, -304, 0], [0, -486, 243, 0], [-1196, 0, 0, 0], [0, -378, 189, 0], [-306, 0, 0, -612], [0, 0, -698, 0], [159, 0, 0, 318], [0, 0, -2612, 0], [2423, 0, 0, 0], [736, 0, 0, 0], [75, 0, 0, 150], [0, 0, 2246, 0], [0, 528, -264, 0], [667, 0, 0, 0], [0, 252, -126, 0], [-441, 0, 0, -882], [276, 0, 0, 552], [0, 6, -3, 0], [1171, 0, 0, 0], [0, 0, -2218, 0], [604, 0, 0, 0], [0, 534, -267, 0], [396, 0, 0, 792], [0, 0, -2288, 0], [12, 0, 0, 24], [521, 0, 0, 0], [0, -588, 294, 0], [0, 0, -316, 0], [249, 0, 0, 498], [0, 696, -348, 0], [-2711, 0, 0, 0], [0, -1020, 510, 0], [-159, 0, 0, -318], [-2566, 0, 0, 0], [0, 0, 1831, 0], [0, -456, 228, 0], [0, 0, -2131, 0], [0, 0, 932, 0], [0, 90, -45, 0], [-458, 0, 0, 0], [0, 0, -1742, 0], [0, -654, 327, 0], [1013, 0, 0, 0], [294, 0, 0, 588], [324, 0, 0, 648], [0, 0, -1909, 0], [0, -360, 180, 0], [-2446, 0, 0, 0], [0, 0, -1289, 0], [0, 0, 2209, 0], [-94, 0, 0, 0], [0, 0, 2078, 0], [0, 0, 1412, 0], [0, 258, -129, 0], [4, 0, 0, 0], [2594, 0, 0, 0], [129, 0, 0, 258], [0, -960, 480, 0], [-2222, 0, 0, 0], [0, -816, 408, 0], [3404, 0, 0, 0], [0, 0, -383, 0], [348, 0, 0, 696], [0, 0, -2822, 0], [0, -1092, 546, 0], [-1012, 0, 0, 0], [12, 0, 0, 24], [72, 0, 0, 144], [0, -336, 168, 0], [0, 1110, -555, 0], [-417, 0, 0, -834], [0, 0, -632, 0], [-138, 0, 0, -276], [-2084, 0, 0, 0], [0, 540, -270, 0], [180, 0, 0, 360], [0, 0, -419, 0], [0, 0, 3349, 0], [0, 960, -480, 0], [-4, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1200_3_c_j(:prec:=4)
    chi := MakeCharacter_1200_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1200_3_c_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1200_3_c_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![64, 0, 1]>,<11,R![315, 0, 1]>],Snew);
    return Vf;
end function;